Sure, let's break this down step-by-step:
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1. Given the function:
[tex]\[ f(x) = \sqrt[3]{8x} + 4 \][/tex]
2. Change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{8x} + 4 \][/tex]
3. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt[3]{8y} + 4 \][/tex]
4. Isolate [tex]\( y \)[/tex]. Start by subtracting 4 from both sides:
[tex]\[ x - 4 = \sqrt[3]{8y} \][/tex]
5. Cube both sides to eliminate the cube root. This results in:
[tex]\[ (x - 4)^3 = 8y \][/tex]
6. Finally, divide both sides by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{(x - 4)^3}{8} \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] can be written as:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]
Hence, the resulting function is:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]
